A Discontinuity in the Distribution of Fixed Point Sums

Details A Discontinuity in the Distribution of Fixed Point Sums A Discontinuity in the Distribution of Fixed Point Sums

2003-04-25

arxiv.org/abs/math/0304416v1

Mathematics

Combinatorics

1 figure

Scientific paper

The quantity $f(n,r)$, defined as the number of permutations of the set $[n]=\{1,2,... n\}$ whose fixed points sum to $r$, shows a sharp discontinuity in the neighborhood of $r=n$. We explain this discontinuity and study the possible existence of other discontinuities in $f(n,r)$ for permutations. We generalize our results to other families of structures that exhibit the same kind of discontinuities, by studying $f(n,r)$ when ``fixed points'' is replaced by ``components of size 1'' in a suitable graph of the structure. Among the objects considered are permutations, all functions and set partitions.

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